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Let $X_1$ and $X_2$ have standard normal distribution and let $(X_1,X_2)$ have a joint bivariate distribution.

Can anyone explain why: $X_1+X_2=\sqrt{2+2\rho}Z$ where $Z\sim N(0,1)$ and $\rho$ is the correlation coefficient?

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I use the fact that any linear combination of normal random variables is a normal random variable, regardless of independence. For a proof, see the answers to this question.

As shown in the second answer to this linked question, by characteristic functions, the variance of $Y = X_1+X_2$ is shown to be $$\sigma^2_Y = \sigma^2_{X_1} + \sigma^2_{X_2} + 2\text{Cov}(X_1, X_2) = \sigma^2_{X_1} + \sigma^2_{X_2} + 2\rho\sigma_{X_1}\sigma_{X_2} = 2 + 2\rho$$ using the formula $\sigma^2_Z=\Sigma_{11}+\Sigma_{22}+2\Sigma_{21}$ given in the answer. Furthermore, it is trivial to see that $\mathbb{E}[X_1 + X_2] = 0$. So, $Y$ follows a normal distribution with mean $0$ and variance $2 + 2\rho$; hence, there exists a $Z \sim \mathcal{N}(0, 1)$ such that $$Z = \dfrac{Y}{\sqrt{2+2\rho}}$$ and the result follows.

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